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Introduction
Understanding Quantum Theory
New Horizons Lectures.
Lecture 13: Understanding Quantum Theory
Philip Goyal
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Objectives
Objectives
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To more sharply articulate what is lacking in our understanding of
quantum theory.
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To introduce the idea of reconstructing quantum theory, and its role in
elucidating the physical meaning of quantum theory.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
What do we lack in our understanding of quantum theory?
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
Classical Physics
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What is it about classical physics that makes it seem understandable?
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Classical physics provides us with a definite conception of physical
reality, the universe as a machine.
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The mathematisation of this conception naturally gives rise to a
mathematical framework — the classical modelling framework.
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Classical physical theories are all built within this framework.
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The details of each theory are drawn from experimental observations
and general physical principles.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
The Mechanical Conception of Physical Reality.
Essential intuition: The physical universe is a vast machine.
By virtue of this conception, the physical universe has some fundamental
properties:
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It is rationally understandable.
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Law-like: It follows quantitative rules (laws) that we can formulate and
comprehend.
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Uniformity of Nature: These rules apply everywhere and at all times.
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Decomposability: It is possible to discover these rules by studying parts of
the universe in isolation from the rest.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
I
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
In particular:
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Universal Time: Everything changes in step with a ‘universal time’.
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Describability: Physical universe completely described at any time by its
state and some intrinsic real parameters.
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Law of Motion: A universal law of motion determines the state at any other
time and is reversible.
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Measurements: Outcomes of measurements are determined by the state.
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Reproducibility: Measurements can be performed in a manner which
is reproducible.
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Transparency: There exists a measurement whose outcomes which
uniquely pin down the state of the system, whatever the state happens to be.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
Classical Modelling Framework
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Time is represented by t ∈ R.
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Physical state is represented by S ∈ S, where S ⊆ Rn .
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Dynamics are represented by one-to-one map, M, over S.
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Measurement is represented by a map, A, from S to Rm .
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Transparency: There exists a reproducible measurement whose
map, A, is invertible and A−1 is onto.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
Newtonian Mechanics
Using experimental observations (Galileo’s laws of motion; Kepler’s laws)
together with intuitive physical notions (force, inertia), a particular theory is
obtained.
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S(t) = (~r1 , . . . , ~rn ; ~ṙ1 , . . . , ~ṙn ) for n particles labelled 1, . . . , n.
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Mt (dt) : S(t) → S(t + dt),
~ 1 dt/m1 , . . . , ~ṙn +F
~ n dt/mn ),
where S(t +dt) = (~r1 +~ṙ1 dt, . . . , ~rn +~ṙn dt; ~ṙ1 +F
~ i is the force on the ith particle and mi is its mass.
where F
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~ i is calculated via Newton’s law of Gravitation.
F
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A reproducible measurement exists which returns the values of the
positions and velocities of all the particles.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
Historical Development of Quantum Theory: 1. A New Mechanics
New mechanics of Schroedinger and Heisenberg: a modification of classical
physics guided by new physical ideas and mathematical guesswork.
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Many non-classical features: statistics, complementarity.
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Many mathematical features (e.g. complex numbers) had no obvious
physical basis or meaning.
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Some key interpretation of the mathematics was given post hoc,
e.g. |ψ(x, t)|2 as probability density and not a physical wave.
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Pre-supposes the existence of the classical, everyday world of
definiteness.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
2. A New Modelling Framework
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Dirac and von Neumann developed a Quantum Modelling Framework
through a mathematically-guided abstraction from the mechanics of
Schroedinger and Heisenberg.
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It inherits from the mechanics of Schroedinger and Heisenberg:
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the mathematical obscurities (complex numbers, unitary transformations)
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the pre-supposition of the existence of a classical, everyday world of
definiteness
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
So, what is lacking in our understanding?
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We lack a clear conception of reality which underpins the quantum
modelling framework.
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In particular, in the quantum modelling framework itself:
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We do not understand the physical basis of many of the key mathematical
features of the formalism.
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We do not understand the presupposition of the existence of a classical
world of definiteness.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
What obstacles stand in our way?
Ideally, we would like a clear conception of reality which, when
mathematised, leads to the quantum modelling framework.
The main obstacles to forming such a conception are:
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Lack of conceptual access to the full mathematical content of the
quantum modelling framework.
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Unclear understanding of the presupposition of the classical world in the
formulation.
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Sheer conceptual difficulty of grasping the non-classical properties of
quantum reality from a single unifying perspective.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
What is needed to overcome these obstacles?
I
We need to know what physical ideas underpin the quantum modelling
framework.
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We need as many clues as possible of what the quantum world ‘looks
like’ (i.e. what properties it has).
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We need the philosophical imagination and philosophical resources to
form a conception of reality that provides a unified understanding of the
various properties
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
What has been done?
1. Interpretations: provide a conception of reality which makes sense of some
features of the quantum modelling framework.
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Copenhagen Interpretation
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Argues for necessity of classicality assumption.
Many Worlds Interpretation
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Attempts to dispense with classicality assumption altogether.
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Argues for a universal wavefunction, and that, via decoherence,
measurements as interactions with all possible outcomes occuring can be
made sense of.
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Argues that probability still makes sense, and that the usual probability rules
of quantum theory can be derived from reasonable assumptions.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
2. Reformulations: mathematically re-express the quantum modelling
framework or quantum mechanics in a new, physically suggestive way.
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Feynman path-integral reformulation
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Quantum mechanics re-expressed in a Lagrangian form.
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A Classical limit of quantum mechanics becomes transparent.
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Powerful aid to intuition in many situations.
Bohm reformulation
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Schroedinger equation re-written in form closely resembling the
Hamilton-Jacobi equation.
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Makes sharply apparent the non-local, instantaneous influences between
particles.
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Inspired Bell’s exploration of non-locality.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
3. Philosophical reflection: develop philosophical understanding of
non-classical features of quantum world.
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Complementarity: Bohr formulated a general philosophical notion of
complementarity.
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Potentiality & Actuality: Heisenberg and Bohm developed the
connections between these Aristotelian notions and complementarity.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
4. Properties: discover properties (especially non-classical) of the quantum
world, and explore their interrelations in general probabilistic frameworks and
toy models.
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no-signalling: in spite of entanglement, it is not possible to use entangled
states to transmit signals instantaneously.
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classical limit behaviour: Ehrenfest’s theorem. Decoherence.
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non-locality: Bell’s theorem.
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no non-contextuality: Kocken-Specker theorem.
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no-cloning: "no universal quantum photocopier" (Wootters and Zurek)
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teleportation: Bennett et al.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
5. Modification: Modification of the mathematical formalism of the quantum
modelling framework.
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Objective Collapse Models
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Penrose’s Gravitationally-Induced objective reduction
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GRW Collapse Model
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory
Introduction
Understanding Quantum Theory
Classical Physics
Historical Development of Quantum Theory
What is lacking in our understanding?
The Obstacles
What has been done?
6. Reconstruction: derive the quantum modelling framework from a set of
physical assumptions.
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Pre-1980s: work of Piron, and others. Logical and abstract.
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Post-1980s: several distinct approaches based on physical ideas. Many
inspired by thinking informationally about quantum theory.
Philip Goyal
New Horizons Lectures. Lecture 13: Understanding Quantum Theory